Problem: $ C = \left[\begin{array}{rr}1 & 4 \\ 4 & -1\end{array}\right]$ $ D = \left[\begin{array}{rrr}3 & -2 & -2 \\ 2 & 3 & 0\end{array}\right]$ What is $ C D$ ?
Answer: Because $ C$ has dimensions $(2\times2)$ and $ D$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C D = \left[\begin{array}{rr}{1} & {4} \\ {4} & {-1}\end{array}\right] \left[\begin{array}{rrr}{3} & \color{#DF0030}{-2} & \color{#9D38BD}{-2} \\ {2} & \color{#DF0030}{3} & \color{#9D38BD}{0}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{3}+{4}\cdot{2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{3}+{4}\cdot{2} & ? & ? \\ {4}\cdot{3}+{-1}\cdot{2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{3}+{4}\cdot{2} & {1}\cdot\color{#DF0030}{-2}+{4}\cdot\color{#DF0030}{3} & ? \\ {4}\cdot{3}+{-1}\cdot{2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{3}+{4}\cdot{2} & {1}\cdot\color{#DF0030}{-2}+{4}\cdot\color{#DF0030}{3} & {1}\cdot\color{#9D38BD}{-2}+{4}\cdot\color{#9D38BD}{0} \\ {4}\cdot{3}+{-1}\cdot{2} & {4}\cdot\color{#DF0030}{-2}+{-1}\cdot\color{#DF0030}{3} & {4}\cdot\color{#9D38BD}{-2}+{-1}\cdot\color{#9D38BD}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}11 & 10 & -2 \\ 10 & -11 & -8\end{array}\right] $